Evaluate _ 0 ^ 1 ^ - 1 x 1 + x ^ 2 dx - Vidyadip

Question

Evaluate $\int _ { 0 } ^ { 1 } \frac { \tan ^ { - 1 } x } { 1 + x ^ { 2 } } dx$

✓

Answer

Let $I = \int _ { 0 } ^ { 1 } \frac { \tan ^ { - 1 } x } { 1 + x ^ { 2 } } d x$
Put $\tan ^ { - 1 } x = t \quad \Rightarrow \frac { 1 } { 1 + x ^ { 2 } } d x = d t$
Lower limit when x = 0, then t = 0
Upper limit when x = 1, then t = $\pi / 4.$
$\therefore I = \int _ { 0 } ^ { \pi / 4 } t d t = \left[ \frac { t ^ { 2 } } { 2 } \right] _ { 0 } ^ { \pi / 4 } = \frac { 1 } { 2 } \left[ \left( \frac { \pi } { 4 } \right) ^ { 2 } - ( 0 ) ^ { 2 } \right] = \frac { \pi ^ { 2 } } { 32 }$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "1 Marks Question" from Integrals→Integrals — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Classify the following as scalar and vector quantities.
Work done.

Differentiate the following w.r.t.x: $\text{e}^{\sin^{-1}\text{x}}$

Write the degree of the differential equation $\frac{\text{d}^{2}\text{y}}{\text{dx}^2}+\text{x}\Big(\frac{\text{dy}}{\text{dx}}\Big)^{2}=2\text{x}^{2}\log\Big(\frac{\text{d}^{2}\text{y}}{\text{dx}^2}\Big).$

The degree of the differential equation $\left(\frac{d^{2} y}{d x^{2}}\right)^{3}+\left(\frac{d y}{d x}\right)^{2}+\sin \left(\frac{d y}{d x}\right)+1=0$ is

Is $x R y, y R z \Rightarrow x R z$ true is transitive relation?

A balloon which always remains spherical, has a variable diameter $\frac{3}{2}\left( {2x + 1} \right)$ Find the rate of change of its volume with respect to x.

Write the value of p for which $\overrightarrow{a} = 3\hat{i} + 2\hat{j} + 9\hat{k}$ and $\overrightarrow{b} = \hat{i} + \text{p}\hat{j} + 3\hat{k}$ are parallel vectors.

Find the values of $a$ and $b$ such that the function defined by $\mathrm{f}(\mathrm{x})= \begin{cases}5, & \text { if } x \leq 2 \\ a x+b, & \text { if } 2 < x < 10 \\ 21, & \text { if } x \geq 10\end{cases}$ is a continuous function.

Show that function $f(x)=x^2, x=0$ is continuous.

Find the position vector of a point which divides the join of points with position vectors $\overrightarrow{\text{a}} - \overrightarrow{\text{2b }} $ and $\overrightarrow{\text{2a}} +\overrightarrow{\text{b}}$externally in the ratio 2:1.